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Applied Econometrics Applied Econometrics Second edition Dimitrios Asteriou and Stephen G. Hall Applied Econometrics HETEROSKEDASTICITY 1. What is Heteroskedasticity 2. Consequences of Heteroskedasticity 3. Detecting Heteroskedasticity 4. Resolving Heteroskedasticity Applied Econometrics Learning Objectives 1. Understand the meaning of heteroskedasticity and homoskedasticity through examples. 2. Understand the consequences of heteroskedasticity on OLS estimates. 3. Detect heteroskedasticity through graph inspection. 4. Detect heteroskedasticity through formal econometric tests. 5. Distinguish among the wide range of available tests for detecting heteroskedasticity. 6. Perform heteroskedasticity tests using econometric software. 7. Resolve heteroskedasticity using econometric software. Applied Econometrics What is Heteroskedasticity Hetero (different or unequal) is the opposite of Homo (same or equal)… Skedastic means spread or scatter… Homoskedasticity = equal spread Heteroskedasticity = unequal spread Applied Econometrics What is Heteroskedasticity Assumption 5 of the CLRM states that the disturbances should have a constant (equal) variance independent of t: Var(ut)=σ2 Therefore, having an equal variance means that the disturbances are homoskedastic. Applied Econometrics What is Heteroskedasticity If the homoskedasticity assumption is violated then Var(ut)=σt2 Where the only difference is the subscript t, attached to the σt2, which means that the variance can change for every different observation in the sample t=1, 2, 3, 4, …, n. Look at the following graphs… Applied Econometrics What is Heteroskedasticity Applied Econometrics What is Heteroskedasticity Applied Econometrics What is Heteroskedasticity Applied Econometrics What is Heteroskedasticity First graph: Homoskedastic residuals Second graph: income-consumption patterns, for low levels of income not much choices, opposite for high levels. Third graph: improvements in data collection techniques (large banks) or to error learning models (experience decreases the chance of making large errors). Applied Econometrics Consequences of Heteroskedasticity 1. The OLS estimators are still unbiased and consistent. This is because none of the explanatory variables is correlated with the error term. So a correctly specified equation will give us values of estimated coefficient which are very close to the real parameters. 2. Affects the distribution of the estimated coefficients increasing the variances of the distributions and therefore making the OLS estimators inefficient. 3. Underestimates the variances of the estimators, leading to higher values of t and F statistics. Applied Econometrics Detecting Heteroskedasticity There are two ways in general. The first is the informal way which is done through graphs and therefore we call it the graphical method. The second is through formal tests for heteroskedasticity, like the following ones: 1. 2. 3. 4. 5. 6. The Breusch-Pagan LM Test The Glesjer LM Test The Harvey-Godfrey LM Test The Park LM Test The Goldfeld-Quandt Tets White’s Test Applied Econometrics Detecting Heteroskedasticity We plot the square of the obtained residuals against fitted Y and the X’s and we see the patterns. Applied Econometrics Detecting Heteroskedasticity Applied Econometrics Detecting Heteroskedasticity Applied Econometrics Detecting Heteroskedasticity Applied Econometrics Detecting Heteroskedasticity Applied Econometrics The Breusch-Pagan LM Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Run the following auxiliary regression: 2 uˆ t a 1 a 2 Z 2 t a 3 Z 3 t ... a p Z pt v t Step 3: Compute LM=nR2, where n and R2 are from the auxiliary regression. Step 4: If LM-stat>χ2p-1 critical reject the null and conclude that there is significant evidence of heteroskedasticity Applied Econometrics The Glesjer LM Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Run the following auxiliary regression: | uˆ t | a 1 a 2 Z 2 t a 3 Z 3 t ... a p Z pt v t Step 3: Compute LM=nR2, where n and R2 are from the auxiliary regression. Step 4: If LM-stat>χ2p-1 critical reject the null and conclude that there is significant evidence of heteroskedasticity Applied Econometrics The Harvey-Godfrey LM Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Run the following auxiliary regression: 2 ˆ ln u t a 1 a 2 Z 2 t a 3 Z 3 t ... a p Z pt v t Step 3: Compute LM=nR2, where n and R2 are from the auxiliary regression. Step 4: If LM-stat>χ2p-1 critical reject the null and conclude that there is significant evidence of heteroskedasticity Applied Econometrics The Park LM Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Run the following auxiliary regression: 2 ln uˆ t a 1 a 2 ln Z 2 t a 3 ln Z 3 t ... a p ln Z pt v t Step 3: Compute LM=nR2, where n and R2 are from the auxiliary regression. Step 4: If LM-stat>χ2p-1 critical reject the null and conclude that there is significant evidence of heteroskedasticity Applied Econometrics The Engle’s ARCH Test Engle introduced a new concept allowing for heteroskedasticity to occur in the variance of the error terms, rather than in the error terms themselves. The key idea is that the variance of ut depends on the size of the squarred error term lagged one period u2t-1 for the first order model or: Var(ut)=γ1+γ2u2t-1 The model can be easily extended for higher orders: Var(ut)=γ1+γ2u2t-1+…+ γpu2t-p Applied Econometrics The Engle’s ARCH Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Regress the squared residuals to a constant and lagged terms of squared residuals, the number of lags will be determined by the hypothesized order of ARCH effects. Step 3: Compute the LM statistic = (n-ρ)R2 from the LM model and compare it with the chi-square critical value. Step 4: Conclude Applied Econometrics The Goldfeld-Quandt Test Step 1: Identify one variable that is closely related to the variance of the disturbances, and order (rank) the observations of this variable in descending order (starting with the highest and going to the lowest). Step 2: Split the ordered sample into two equally sized sub-samples by omitting c central observations, so that the two samples will contain ½(n-c) observations. Applied Econometrics The Goldfeld-Quandt Test Step 3:Run and OLS regression of Y on the X variable that you have used in step 1 for each subsample and obtain the RSS for each equation. Step 4: Caclulate the F-stat=RSS1/RSS2, where RSS1 is the RSS with the largest value. Step 5: If F-stat>F-crit(1/2(n-c)-l,1/2(n-c)-k) reject the null of homoskedasticity. Applied Econometrics The White’s Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Run the following auxiliary regression: 2 2 2 uˆ t a1 a 2 X 2 t a 3 X 3 t a 4 X 2 t a 5 X 3 t a 6 X 2 t X 3 t v t Step 3: Compute LM=nR2, where n and R2 are from the auxiliary regression. Step 4: If LM-stat>χ2p-1 critical reject the null and conclude that there is significant evidence of heteroskedasticity Applied Econometrics Resolving Heteroskedasticity We have three different cases: (a) Generalized Least Squares (b) Weighted Least Squares (c) Heteroskedasticity-Consistent Estimation Methods Applied Econometrics Generalized Least Squares Consider the model Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut where Var(ut)=σt2 Applied Econometrics Generalized Least Squares If we divide each term by the standard deviation of the error term, σt we get: Yt=β1 (1/σt) +β2X2t/σt +β3X3t/σt +…+βkXkt/σt +ut/σt or Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+u*t Where we have now that: Var(u*t)=Var(ut/σt)=Var(ut)/σt2=1 Applied Econometrics Weighted Least Squares The GLS procedure is the same as the WLS where we have weights, wt, adjusting our variables. Define wt=1/σt, and rewrite the original model as: wtYt=β1wt+β2X2twt+β3X3twt+…+βkXktwt+utwt Where if we define as wtYt-1=Y*t and Xitwt=X*it we get Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+u*t